document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=287659>Question 287659</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If you have a peice of paper that is 84\" x 84\". You are trying to make a box our of it and cut our squares in each corner. What is the maximum volume the box can be? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #208416 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=208416\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Assume you have a square of paper/cardboard/whatever of dimension <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\">.  If you cut squares of dimension <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> from each of the four corners, then the square base of your box will have dimension <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ -\ 2x\">, and the height of the box will be simply <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.\r<BR>\n" );
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document.write( "Therefore the volume of the box will be:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V(x)\ =\ x(a\ -\ 2x)^2 = 4x^3\ -\ 4ax^2\ +\ a^2x\">\r<BR>\n" );
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document.write( "Take the first derivative:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dV}{dx}\ =\ 12x^2\ -\ 8ax\ +\ a^2\">\r<BR>\n" );
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document.write( "Which we need to set equal to zero and solve to find a local extremum:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 12x^2\ -\ 8ax\ +\ a^2\ =\ 0\">\r<BR>\n" );
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document.write( "Since for this problem we know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> is a multiple of 12 we can simplify further by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2\ -\ \frac{8a}{12}x\ +\ \frac{a^2}{12}\ =\ 0\">\r<BR>\n" );
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document.write( "Now substitute the given 84\" for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2\ -\ 56x\ +\ 588\ =\ 0\">\r<BR>\n" );
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document.write( "Which factors to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ -\ 42)(x\ -\ 14)\ =\ 0\">\r<BR>\n" );
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document.write( "Hence the roots and potential extrema are at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 42\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 14\">.  Quite obviously we can exclude <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 42\"> in looking for maximum volume because if you cut 42\" squares from the corners of an 84\" by 84\" sheet of paper, you won't have anything left for the bottom of the box.\r<BR>\n" );
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document.write( "Take the second derivative\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2V}{dx^2}\ =\ 24x\ -\ 8a\ =\ 24x\ -\ 672\">\r<BR>\n" );
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document.write( "Now evaluate the second derivative at the extreme point:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V''(14)\ =\ 24(14)\ -\ 672\ <\ 0\">\r<BR>\n" );
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document.write( "A negative second derivative at a local extremum indicates a local maximum.\r<BR>\n" );
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document.write( "Therefore the cutout for the maximum volume is 14\".\r<BR>\n" );
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document.write( "The volume of the box when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 14\"> is given by the volume function evaluated at 14:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ V(14)\ = 4(14)^3\ -\ 4(84)(14)^2\ +\ (84)^2(14)\"> \r<BR>\n" );
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document.write( "You can do your own arithmetic.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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